I am using the API key I generated for our Hubspot CRM account. I was told by support that the issue will be resolved by going through the Oauth2 flow, though I’m not sure how that would help. How can I get a developer API key?

The endpoint you mentioned in your original post is the Create object type endpoint (see below), which is used to create an object type that your CRM extension to use. Because object types are tied to your app, not your portal, you need to use a developer API key from the developer portal your app is hosted. You can get this key by logging into your developer portal and clicking the Get HAPIkey button on the top right.

Yes, I'm getting the same error:
403 {"status":"error","message":"This hapikey (HAPAIKEY) does not have proper permissions! (requires all of [developers-access])","correlationId":"DHTFHHGFFGBVF","requestId":"c5566dhdjjdg7788"}